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Beliebt Rechnen >

integral of (5x^2-4x)e^{2x}

Lösung

\int (5 x^{2} - 4 x) e^{2 x} d x

Lösung

\frac{1}{4} e^{2 x} (10 x^{2} - 18 x + 9) + C

Schritte zur Lösung

\int (5 x^{2} - 4 x) e^{2 x} d x

Wende U-Substitution an

= \int \frac{5 ln ^{2} ( u ) - 8 ln ( u )}{8} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{8} \cdot \int 5 ln^{2} (u) - 8 ln (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{8} (\int 5 ln^{2} (u) d u - \int 8 ln (u) d u)

\int 5 ln^{2} (u) d u = 5 (u ln^{2} (u) - 2 (u ln (u) - u))

\int 8 ln (u) d u = 8 (u ln (u) - u)

= \frac{1}{8} (5 (u ln^{2} (u) - 2 (u ln (u) - u)) - 8 (u ln (u) - u))

Setze in

u = e^{2 x}

ein

= \frac{1}{8} (5 (e^{2 x} ln^{2} (e^{2 x}) - 2 (e^{2 x} ln (e^{2 x}) - e^{2 x})) - 8 (e^{2 x} ln (e^{2 x}) - e^{2 x}))

Vereinfache

\frac{1}{8} (5 (e^{2 x} ln^{2} (e^{2 x}) - 2 (e^{2 x} ln (e^{2 x}) - e^{2 x})) - 8 (e^{2 x} ln (e^{2 x}) - e^{2 x})) : \frac{1}{4} e^{2 x} (10 x^{2} - 18 x + 9)

= \frac{1}{4} e^{2 x} (10 x^{2} - 18 x + 9)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} e^{2 x} (10 x^{2} - 18 x + 9) + C

Graph

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